# Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

arXiv: https://arxiv.org/abs/2307.10466 · HTML: https://mlsanity.com/paper/2307.10466/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2023.

## Built on by (most cited first, 3 of 3)

- [Learning the Sherrington-Kirkpatrick Model Even at Low Temperature](https://mlsanity.com/paper/2411.11174/) (2024)
- [Estimating Ising Models in Total Variation Distance](https://mlsanity.com/paper/2511.21008/) (2025)
- [On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy](https://mlsanity.com/paper/2606.12720/) (2026)

## Abstract (arXiv)

We study Glauber dynamics for sampling from discrete distributions μ on the hypercube ± 1n. Recently, techniques based on spectral independence have successfully yielded optimal O(n) relaxation times for a host of different distributions μ. We show that spectral independence is universal: a relaxation time of O(n) implies spectral independence. We then study a notion of tractability for μ, defined in terms of smoothness of the multilinear extension of its Hamiltonian -- log μ -- over [-1,+1]n. We show that Glauber dynamics has relaxation time O(n) for such μ, and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal widetildeO(n) mixing time for random Hamiltonians, i.e. the classically studied mixed p-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.